The multinomial theorem states that $$ \sum_{\substack{n_1 \geq 0, \ldots, n_k \geq 0\\ n_1 + \cdots + n_k = n}} {n \choose n_1, \ldots, n_k} \, p_1^{n_1} \cdots p_k^{n_k} = (p_1 + \cdots + p_k)^n \,. $$
Is there a simple closed-form expression for the closely-related sum $$ \sum_{\substack{m > n_1 \geq 0, \ldots, m > n_k \ge 0\\ n_1 + \cdots + n_k = n}} {n \choose n_1, \ldots, n_k} \, p_1^{n_1} \cdots p_k^{n_k} \,, $$ where no summed-over integer is allowed to exceed some maximum value, in this case $m$?
Evaluating this sum would be useful in calculating the probability that a given event $i \in \{1, \ldots, k\}$ is the mode of a sample of a multinomial distribution.